Don founded Kamakura Corporation in April 1990 and currently serves as its chairman and chief executive officer where he focuses on enterprise wide risk management and modern credit risk technology. His primary financial consulting and research interests involve the practical application of leading edge financial theory to solve critical financial risk management problems. Don was elected to the 50 member RISK Magazine Hall of Fame in 2002 for his work at Kamakura. Read More

In this note, we address the same three questions that we raised in the March 2015 piece, but this time the answers are derived for the Government of Canada yield curve:

How do you measure the accuracy of an interest rate risk simulation technique?

Given that measure of accuracy, how many risk factors are necessary?

How does accuracy change as the number of factors increases?

In answering the question “how good is good enough” for interest rate risk modeling, we follow the procedures that Bharath and Shumway (2008) used in testing the accuracy of the Merton model of risky debt versus the reduced form approach to credit risk modeling. We test these two hypotheses about one factor term structure models:

Strong form of hypothesis:One factor term structure models are so accurate that there are no other variables than the first factor that have statistically significant explanatory power.

Weaker form of hypothesis:There are other factors beyond the first factor that are statistically significant, but their impact is very modest and the benefits of using more than one factor are very minor.

Non-Parametric Tests of One Factor Term Structure Models
Jarrow, van Deventer and Wang (2003) (“JvDW”) provide another testing procedure that we address first. In examining the Merton model of risky debt, JvDW provide a very intuitive testing procedure that is independent of the parameters fitted to the model structure. They asked this question: “Are the implications of the model true or false?” Since no model is perfect, they answer this question with a probability.

We address two classes of one factor term structure models in this section:

Non-parametric test 1: The one factor models with rate-dependent interest rate volatility make it impossible for interest rates to be negative. Is this implication true or false? It is false. While rates for the Government of Canada have not yet been negative, negative rates are so common in Europe, Japan and Hong Kong that this class of one factor model should be rejected as unsuitable on its face. As confirmation, here is the recent history of interest rates in Switzerland provided by the Swiss National Bank:

Non-parametric test 2:The Vasicek, Ho and Lee, and Extended Vasicek/Hull and White models assume that interest rate volatility is a constant, independent of the level of interest rates. This assumption implies that both the level and the changes in interest rates are normally distributed over time. We use quarterly data on Government of Canada yields provided by the Bank of Canada and Bloomberg for the period from July 1, 1997 through March 31, 2015. We extract zero coupon bond yields from this data using Kamakura Risk Manager version 8.1 and maximum smoothness forward rate smoothing. This graph shows the evolution of Government of Canada zero coupon yields over time:

This graph shows the evolution of the first quarterly forward rate (the forward that applies from the 91st day through the 182nd day) over the same time period:

We use three statistical tests to determine whether or not the hypothesis of normality should be rejected at the 5% level for two sets of data:

The absolute level of zero coupon bond yields over the 1997 to 2015 time period

The quarterly changes in 120 quarterly forward rates making up the 30 year Government of Canada yield curve.

The chart above shows the p-values for these three statistical tests for the first twelve quarterly maturities. The null hypothesis of normality is rejected by all 3 tests for all of the 120 quarterly zero coupon yield maturities. For quarterly changes in forward rates, the null hypothesis of normality is rejected by all 3 tests for 75 of the first 86 maturities. Non-parametric test 3:As commonly implemented, one factor term structure models imply that all yields will either (a) rise, (b) fall, or (c) remain unchanged. In Chapter 3 of Advanced Financial Management (second edition, 2013), van Deventer, Imai and Mesler show that this implication of one factor term structure models is rarely true in the U.S. Treasury market. We perform the same test using 4,689 days of zero coupon bond yields for the Government of Canada yield curve. We analyze the daily shifts in the 360 different monthly zero coupon bond yields on each day. The results are given here:

The results were rarely consistent with the implications of one factor term structure models. Yield curve shifts were all positive, all negative, or all zero 7.61%, 8.94%, and 0% of the time, a total of 16.65% of all business days. The predominant yield curve shift was a twist, with a mix of positive changes, negative changes, or zero changes. These twists, which happen 83.45% of the time in Canada, cannot be modeled at all with one factor term structure models. Non-parametric test 4: A closely related test is discussed in Chapter 3 of van Deventer, Imai and Mesler. One factor term structure models cannot create a yield curve that has multiple humps in it. One simply has to count the humps in the Government of Canada yield curve to show that this is another serious problem with one factor term structure models:

The number of days with 0 or 1 humps (defined as the sum of local minima and maxima on that day’s yield curve) was only 8.76% of the total of 4,690 days in the data. The remainder of the data set, 91.24% of the total, has yield curves with shapes that are inconsistent with a one factor term structure model.

Fitting a Multi-Factor Heath Jarrow and Morton Term Structure Model to Government of Canada Yields
Given the poor performance of one factor models on the non-parametric tests above, it is no surprise that bank regulators are turning to multi-factor models around the world. The Federal Reserve’s Comprehensive Capital Analysis and Review stress testing regime has included 3 points on the U.S. Treasury yield curve since 2014. The Bank for International Settlements has required that at least six interest rate risk factors be used to model market risk since 2010.

We now fit a multi-factor Heath, Jarrow and Morton model to quarterly Government of Canada yield data from July 1, 1997 to March 31, 2015. The same yields were available for all observations: 3 months, 6 months, and 1, 2, 3, 4, 5, 7, 10, 20 and 30 years. The procedures used are described in detail in these documents:

We extract the zero coupon yields and zero coupon bond prices for all quarterly maturities out to thirty years for all daily observations. This is done using Kamakura Risk Manager, version 8.1, using the maximum smoothness forward rate approach.

We drop the daily observations that are not the last observation of the quarter, to avoid overlapping quarterly observations and the resulting autocorrelated errors that would stem from that.

We calculate the continuously compounded changes in forward returns as described in the parameter technical guide.

We then begin the process of creating the orthogonalized risk factors that drive interest rates. These factors are assumed to be uncorrelated independent random variables that have a normal distribution with mean zero and standard deviation of 1.

In the estimation process, we added factors to the model as long as each new factor provided incremental explanatory power.

We use the resulting parameters and accuracy tests to address the hypothesis that a one factor model is “good enough” for modeling Government of Canada yields.

Proof That One Factor Models Are Not Sufficient for Best Practice Risk Management
We now test the hypotheses about one factor term structure models.

Strong form of hypothesis: One factor term structure models are so accurate that there are no other variables than the first factor that have statistically significant explanatory power.

The following graph shows that a one factor term structure model omits a very large number of risk factors driving Government of Canada yields:

Other than the first quarterly forward rate in the yield curve, there are at least 12 explanatory variables that drive the 120 quarterly segments of the yield curve. The final Government of Canada term structure model from Kamakura Risk Information services has 12 independent risk factors that drive yields, and these factors also appear in combination with rate level variables. A total of 27 related candidate explanatory variables were used in the estimation process.

Conclusion:
The strong form of the hypothesis is overwhelmingly rejected by the data on Government of Canada yields.

We now turn to the weaker hypothesis.

Weaker form of hypothesis:There are other factors beyond the first factor that are statistically significant, but their impact is very modest and the benefits of using more than one factor are very minor.

To address this hypothesis, we graph the adjusted r-squared of a “regime change” one factor model which combines normally distributed and rate dependent one factor models with a best practice model which includes all statistically significant factors. The results are shown here:

The adjusted r-squared for the best practice model is plotted in blue and is near 100% for all 120 quarterly segments of the yield curve. The one factor model, by contrast, does a stunningly poor job of fitting quarterly movements in the quarterly forward rates. The adjusted r-squared is good, of course, for the first forward rate since the short rate is the standard risk factor in a one factor term structure model. Beyond the first quarter, however, explanatory power is extremely low. The adjusted r-squared of the one factor model never exceeds 35% and is far below that level at most maturities.

This result should not come as a surprise to a serious analyst, because it is very similar to the results of the best practice Heath, Jarrow and Morton term structure model for U.S. Treasuries.

We can confirm the low explanatory power of a one factor model with a one line principal components analysis in a common statistical package. The “PCA” analysis is not constrained to choose the short rate as the explanatory variable in a one factor model. In fact, there are many other factors that would be stronger candidates for a single factor model. The results of the principal components analysis on quarterly movements in Government of Canada yields are shown here:

The results show that at least 10 factors are needed to model the Government of Canada yield curve with cumulative accuracy comparable to the confidence levels most large financial institutions would use for value at risk analysis. The first factor explains only 57% of quarterly forward rate movements.

Conclusion:
The weak form of the hypothesis is also overwhelmingly rejected by the data on Government of Canada yields.

Can a One Factor Model be “Tweaked” with One or Two More Factors?Many large financial institutions have been using one factor models for such a long time that hope springs eternal that they can be fixed with a small “tweak,” a second or third factor. In the next graph we show the adjusted r-squareds for 1, 2, 3, 6 and “all” factors in a model of the Government of Canada yield curve.

The results show that even a six factor model leaves a big chunk of yield curve movements unexplained. In the 21st century, with modern big data technology, using all factors that matter, instead of just a few of them, is a simple step forward.

A similar plot of the root mean squared errors for 1 factor, 2 factor, 3 factor, 6 factor, and all factor term structure models shows the danger of half steps in improving interest rate risk technology:

Research subscriptions to Heath, Jarrow and Morton term structure modelingThis is a good first step for regulatory agencies and financial institutions building their familiarity with modern interest rate risk technology. The subscription includes the Technical Guides describing the parameter estimation process, the underlying raw data, and the parameters themselves, updated annually. Models are available for all major government yield curves.

Production subscription to Heath, Jarrow and Morton term structure modelingThe production subscription includes formatting of parameters for use in Kamakura Risk Manager’s newest versions and immediate release of HJM parameter estimates as soon as the quality control process at Kamakura Corporation is completed.

Production subscription to HJM yield scenarios
Kamakura Risk Information Services also generates the scenarios in-house and provides the scenarios in standard Kamakura Risk Manager format for all major government yield curves at customized frequencies (daily, weekly, monthly, quarterly) for individual clients. Transfer of data is by file transfer protocol technology.

Heath, Jarrow and Morton Training
Kamakura Corporation, led by Managing Director Robert A. Jarrow (Cornell University) provides training in modern Heath, Jarrow and Morton interest rate risk technology for both clients and potential clients. Professor Jarrow usually participates by video link in these training sessions.

For inquiries about these and other products, please contact your Kamakura representative or e-mail Kamakura at info@kamakuraco.com.

Further Reading for the Technically Inclined Reader

References for random interest rate modeling are given here:

Heath, David, Robert A. Jarrow and Andrew Morton, "Bond Pricing and the Term Structure of Interest Rates: A Discrete Time Approach," Journal of Financial and Quantitative Analysis, 1990, pp. 419-440.